Properties

Label 6046.2.a.f.1.12
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $67$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.14148 q^{3} +1.00000 q^{4} +3.02625 q^{5} -2.14148 q^{6} +4.90651 q^{7} +1.00000 q^{8} +1.58595 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.14148 q^{3} +1.00000 q^{4} +3.02625 q^{5} -2.14148 q^{6} +4.90651 q^{7} +1.00000 q^{8} +1.58595 q^{9} +3.02625 q^{10} +2.36471 q^{11} -2.14148 q^{12} +2.98785 q^{13} +4.90651 q^{14} -6.48067 q^{15} +1.00000 q^{16} +0.102887 q^{17} +1.58595 q^{18} +0.460748 q^{19} +3.02625 q^{20} -10.5072 q^{21} +2.36471 q^{22} +0.572097 q^{23} -2.14148 q^{24} +4.15821 q^{25} +2.98785 q^{26} +3.02817 q^{27} +4.90651 q^{28} -4.49268 q^{29} -6.48067 q^{30} -1.20673 q^{31} +1.00000 q^{32} -5.06398 q^{33} +0.102887 q^{34} +14.8483 q^{35} +1.58595 q^{36} -1.90370 q^{37} +0.460748 q^{38} -6.39843 q^{39} +3.02625 q^{40} +11.7432 q^{41} -10.5072 q^{42} +7.15540 q^{43} +2.36471 q^{44} +4.79948 q^{45} +0.572097 q^{46} -2.47929 q^{47} -2.14148 q^{48} +17.0738 q^{49} +4.15821 q^{50} -0.220331 q^{51} +2.98785 q^{52} +6.38183 q^{53} +3.02817 q^{54} +7.15620 q^{55} +4.90651 q^{56} -0.986685 q^{57} -4.49268 q^{58} -6.28216 q^{59} -6.48067 q^{60} -10.0775 q^{61} -1.20673 q^{62} +7.78147 q^{63} +1.00000 q^{64} +9.04198 q^{65} -5.06398 q^{66} -0.866688 q^{67} +0.102887 q^{68} -1.22514 q^{69} +14.8483 q^{70} -10.2469 q^{71} +1.58595 q^{72} +0.484504 q^{73} -1.90370 q^{74} -8.90473 q^{75} +0.460748 q^{76} +11.6025 q^{77} -6.39843 q^{78} -1.79064 q^{79} +3.02625 q^{80} -11.2426 q^{81} +11.7432 q^{82} -8.29759 q^{83} -10.5072 q^{84} +0.311363 q^{85} +7.15540 q^{86} +9.62100 q^{87} +2.36471 q^{88} +6.42816 q^{89} +4.79948 q^{90} +14.6599 q^{91} +0.572097 q^{92} +2.58419 q^{93} -2.47929 q^{94} +1.39434 q^{95} -2.14148 q^{96} -3.20410 q^{97} +17.0738 q^{98} +3.75031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.14148 −1.23639 −0.618193 0.786026i \(-0.712135\pi\)
−0.618193 + 0.786026i \(0.712135\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.02625 1.35338 0.676691 0.736267i \(-0.263413\pi\)
0.676691 + 0.736267i \(0.263413\pi\)
\(6\) −2.14148 −0.874257
\(7\) 4.90651 1.85448 0.927242 0.374462i \(-0.122173\pi\)
0.927242 + 0.374462i \(0.122173\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.58595 0.528650
\(10\) 3.02625 0.956985
\(11\) 2.36471 0.712986 0.356493 0.934298i \(-0.383972\pi\)
0.356493 + 0.934298i \(0.383972\pi\)
\(12\) −2.14148 −0.618193
\(13\) 2.98785 0.828680 0.414340 0.910122i \(-0.364012\pi\)
0.414340 + 0.910122i \(0.364012\pi\)
\(14\) 4.90651 1.31132
\(15\) −6.48067 −1.67330
\(16\) 1.00000 0.250000
\(17\) 0.102887 0.0249538 0.0124769 0.999922i \(-0.496028\pi\)
0.0124769 + 0.999922i \(0.496028\pi\)
\(18\) 1.58595 0.373812
\(19\) 0.460748 0.105703 0.0528514 0.998602i \(-0.483169\pi\)
0.0528514 + 0.998602i \(0.483169\pi\)
\(20\) 3.02625 0.676691
\(21\) −10.5072 −2.29286
\(22\) 2.36471 0.504157
\(23\) 0.572097 0.119290 0.0596452 0.998220i \(-0.481003\pi\)
0.0596452 + 0.998220i \(0.481003\pi\)
\(24\) −2.14148 −0.437128
\(25\) 4.15821 0.831641
\(26\) 2.98785 0.585965
\(27\) 3.02817 0.582771
\(28\) 4.90651 0.927242
\(29\) −4.49268 −0.834270 −0.417135 0.908844i \(-0.636966\pi\)
−0.417135 + 0.908844i \(0.636966\pi\)
\(30\) −6.48067 −1.18320
\(31\) −1.20673 −0.216735 −0.108367 0.994111i \(-0.534562\pi\)
−0.108367 + 0.994111i \(0.534562\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.06398 −0.881526
\(34\) 0.102887 0.0176450
\(35\) 14.8483 2.50982
\(36\) 1.58595 0.264325
\(37\) −1.90370 −0.312966 −0.156483 0.987681i \(-0.550016\pi\)
−0.156483 + 0.987681i \(0.550016\pi\)
\(38\) 0.460748 0.0747432
\(39\) −6.39843 −1.02457
\(40\) 3.02625 0.478493
\(41\) 11.7432 1.83398 0.916988 0.398916i \(-0.130613\pi\)
0.916988 + 0.398916i \(0.130613\pi\)
\(42\) −10.5072 −1.62130
\(43\) 7.15540 1.09119 0.545594 0.838050i \(-0.316304\pi\)
0.545594 + 0.838050i \(0.316304\pi\)
\(44\) 2.36471 0.356493
\(45\) 4.79948 0.715465
\(46\) 0.572097 0.0843511
\(47\) −2.47929 −0.361642 −0.180821 0.983516i \(-0.557875\pi\)
−0.180821 + 0.983516i \(0.557875\pi\)
\(48\) −2.14148 −0.309096
\(49\) 17.0738 2.43911
\(50\) 4.15821 0.588059
\(51\) −0.220331 −0.0308525
\(52\) 2.98785 0.414340
\(53\) 6.38183 0.876612 0.438306 0.898826i \(-0.355579\pi\)
0.438306 + 0.898826i \(0.355579\pi\)
\(54\) 3.02817 0.412081
\(55\) 7.15620 0.964942
\(56\) 4.90651 0.655659
\(57\) −0.986685 −0.130690
\(58\) −4.49268 −0.589918
\(59\) −6.28216 −0.817867 −0.408934 0.912564i \(-0.634099\pi\)
−0.408934 + 0.912564i \(0.634099\pi\)
\(60\) −6.48067 −0.836651
\(61\) −10.0775 −1.29030 −0.645148 0.764058i \(-0.723204\pi\)
−0.645148 + 0.764058i \(0.723204\pi\)
\(62\) −1.20673 −0.153255
\(63\) 7.78147 0.980373
\(64\) 1.00000 0.125000
\(65\) 9.04198 1.12152
\(66\) −5.06398 −0.623333
\(67\) −0.866688 −0.105883 −0.0529414 0.998598i \(-0.516860\pi\)
−0.0529414 + 0.998598i \(0.516860\pi\)
\(68\) 0.102887 0.0124769
\(69\) −1.22514 −0.147489
\(70\) 14.8483 1.77471
\(71\) −10.2469 −1.21609 −0.608045 0.793903i \(-0.708046\pi\)
−0.608045 + 0.793903i \(0.708046\pi\)
\(72\) 1.58595 0.186906
\(73\) 0.484504 0.0567070 0.0283535 0.999598i \(-0.490974\pi\)
0.0283535 + 0.999598i \(0.490974\pi\)
\(74\) −1.90370 −0.221300
\(75\) −8.90473 −1.02823
\(76\) 0.460748 0.0528514
\(77\) 11.6025 1.32222
\(78\) −6.39843 −0.724479
\(79\) −1.79064 −0.201463 −0.100732 0.994914i \(-0.532118\pi\)
−0.100732 + 0.994914i \(0.532118\pi\)
\(80\) 3.02625 0.338345
\(81\) −11.2426 −1.24918
\(82\) 11.7432 1.29682
\(83\) −8.29759 −0.910779 −0.455389 0.890292i \(-0.650500\pi\)
−0.455389 + 0.890292i \(0.650500\pi\)
\(84\) −10.5072 −1.14643
\(85\) 0.311363 0.0337720
\(86\) 7.15540 0.771586
\(87\) 9.62100 1.03148
\(88\) 2.36471 0.252079
\(89\) 6.42816 0.681384 0.340692 0.940175i \(-0.389339\pi\)
0.340692 + 0.940175i \(0.389339\pi\)
\(90\) 4.79948 0.505910
\(91\) 14.6599 1.53677
\(92\) 0.572097 0.0596452
\(93\) 2.58419 0.267968
\(94\) −2.47929 −0.255719
\(95\) 1.39434 0.143056
\(96\) −2.14148 −0.218564
\(97\) −3.20410 −0.325327 −0.162664 0.986682i \(-0.552009\pi\)
−0.162664 + 0.986682i \(0.552009\pi\)
\(98\) 17.0738 1.72471
\(99\) 3.75031 0.376920
\(100\) 4.15821 0.415821
\(101\) −16.0270 −1.59475 −0.797373 0.603487i \(-0.793777\pi\)
−0.797373 + 0.603487i \(0.793777\pi\)
\(102\) −0.220331 −0.0218160
\(103\) −12.1875 −1.20087 −0.600434 0.799674i \(-0.705005\pi\)
−0.600434 + 0.799674i \(0.705005\pi\)
\(104\) 2.98785 0.292983
\(105\) −31.7974 −3.10311
\(106\) 6.38183 0.619858
\(107\) 16.0032 1.54708 0.773542 0.633745i \(-0.218483\pi\)
0.773542 + 0.633745i \(0.218483\pi\)
\(108\) 3.02817 0.291385
\(109\) −12.5104 −1.19828 −0.599142 0.800643i \(-0.704491\pi\)
−0.599142 + 0.800643i \(0.704491\pi\)
\(110\) 7.15620 0.682317
\(111\) 4.07674 0.386947
\(112\) 4.90651 0.463621
\(113\) −6.12533 −0.576222 −0.288111 0.957597i \(-0.593027\pi\)
−0.288111 + 0.957597i \(0.593027\pi\)
\(114\) −0.986685 −0.0924115
\(115\) 1.73131 0.161446
\(116\) −4.49268 −0.417135
\(117\) 4.73858 0.438081
\(118\) −6.28216 −0.578319
\(119\) 0.504816 0.0462764
\(120\) −6.48067 −0.591601
\(121\) −5.40816 −0.491651
\(122\) −10.0775 −0.912377
\(123\) −25.1478 −2.26750
\(124\) −1.20673 −0.108367
\(125\) −2.54748 −0.227854
\(126\) 7.78147 0.693228
\(127\) −3.67448 −0.326058 −0.163029 0.986621i \(-0.552126\pi\)
−0.163029 + 0.986621i \(0.552126\pi\)
\(128\) 1.00000 0.0883883
\(129\) −15.3232 −1.34913
\(130\) 9.04198 0.793034
\(131\) 11.9430 1.04347 0.521733 0.853109i \(-0.325286\pi\)
0.521733 + 0.853109i \(0.325286\pi\)
\(132\) −5.06398 −0.440763
\(133\) 2.26066 0.196024
\(134\) −0.866688 −0.0748704
\(135\) 9.16399 0.788711
\(136\) 0.102887 0.00882250
\(137\) −15.3823 −1.31420 −0.657101 0.753803i \(-0.728217\pi\)
−0.657101 + 0.753803i \(0.728217\pi\)
\(138\) −1.22514 −0.104291
\(139\) −17.3330 −1.47017 −0.735083 0.677977i \(-0.762857\pi\)
−0.735083 + 0.677977i \(0.762857\pi\)
\(140\) 14.8483 1.25491
\(141\) 5.30936 0.447129
\(142\) −10.2469 −0.859905
\(143\) 7.06539 0.590837
\(144\) 1.58595 0.132162
\(145\) −13.5960 −1.12909
\(146\) 0.484504 0.0400979
\(147\) −36.5632 −3.01569
\(148\) −1.90370 −0.156483
\(149\) 2.21301 0.181297 0.0906483 0.995883i \(-0.471106\pi\)
0.0906483 + 0.995883i \(0.471106\pi\)
\(150\) −8.90473 −0.727068
\(151\) 14.7036 1.19656 0.598282 0.801286i \(-0.295850\pi\)
0.598282 + 0.801286i \(0.295850\pi\)
\(152\) 0.460748 0.0373716
\(153\) 0.163174 0.0131918
\(154\) 11.6025 0.934952
\(155\) −3.65187 −0.293325
\(156\) −6.39843 −0.512284
\(157\) 11.5889 0.924897 0.462449 0.886646i \(-0.346971\pi\)
0.462449 + 0.886646i \(0.346971\pi\)
\(158\) −1.79064 −0.142456
\(159\) −13.6666 −1.08383
\(160\) 3.02625 0.239246
\(161\) 2.80700 0.221222
\(162\) −11.2426 −0.883303
\(163\) 4.95134 0.387819 0.193909 0.981019i \(-0.437883\pi\)
0.193909 + 0.981019i \(0.437883\pi\)
\(164\) 11.7432 0.916988
\(165\) −15.3249 −1.19304
\(166\) −8.29759 −0.644018
\(167\) 7.93708 0.614190 0.307095 0.951679i \(-0.400643\pi\)
0.307095 + 0.951679i \(0.400643\pi\)
\(168\) −10.5072 −0.810648
\(169\) −4.07277 −0.313290
\(170\) 0.311363 0.0238804
\(171\) 0.730723 0.0558798
\(172\) 7.15540 0.545594
\(173\) 10.4968 0.798055 0.399028 0.916939i \(-0.369348\pi\)
0.399028 + 0.916939i \(0.369348\pi\)
\(174\) 9.62100 0.729366
\(175\) 20.4023 1.54227
\(176\) 2.36471 0.178247
\(177\) 13.4531 1.01120
\(178\) 6.42816 0.481811
\(179\) −3.74264 −0.279738 −0.139869 0.990170i \(-0.544668\pi\)
−0.139869 + 0.990170i \(0.544668\pi\)
\(180\) 4.79948 0.357732
\(181\) 10.2544 0.762201 0.381100 0.924534i \(-0.375545\pi\)
0.381100 + 0.924534i \(0.375545\pi\)
\(182\) 14.6599 1.08666
\(183\) 21.5809 1.59530
\(184\) 0.572097 0.0421756
\(185\) −5.76107 −0.423562
\(186\) 2.58419 0.189482
\(187\) 0.243298 0.0177917
\(188\) −2.47929 −0.180821
\(189\) 14.8577 1.08074
\(190\) 1.39434 0.101156
\(191\) −13.3846 −0.968474 −0.484237 0.874937i \(-0.660903\pi\)
−0.484237 + 0.874937i \(0.660903\pi\)
\(192\) −2.14148 −0.154548
\(193\) −13.0787 −0.941427 −0.470713 0.882286i \(-0.656003\pi\)
−0.470713 + 0.882286i \(0.656003\pi\)
\(194\) −3.20410 −0.230041
\(195\) −19.3633 −1.38663
\(196\) 17.0738 1.21956
\(197\) 0.464823 0.0331173 0.0165586 0.999863i \(-0.494729\pi\)
0.0165586 + 0.999863i \(0.494729\pi\)
\(198\) 3.75031 0.266523
\(199\) −15.2055 −1.07789 −0.538943 0.842342i \(-0.681176\pi\)
−0.538943 + 0.842342i \(0.681176\pi\)
\(200\) 4.15821 0.294030
\(201\) 1.85600 0.130912
\(202\) −16.0270 −1.12766
\(203\) −22.0434 −1.54714
\(204\) −0.220331 −0.0154263
\(205\) 35.5378 2.48207
\(206\) −12.1875 −0.849142
\(207\) 0.907317 0.0630629
\(208\) 2.98785 0.207170
\(209\) 1.08953 0.0753647
\(210\) −31.7974 −2.19423
\(211\) −7.19882 −0.495587 −0.247793 0.968813i \(-0.579705\pi\)
−0.247793 + 0.968813i \(0.579705\pi\)
\(212\) 6.38183 0.438306
\(213\) 21.9437 1.50356
\(214\) 16.0032 1.09395
\(215\) 21.6540 1.47679
\(216\) 3.02817 0.206041
\(217\) −5.92082 −0.401932
\(218\) −12.5104 −0.847314
\(219\) −1.03756 −0.0701117
\(220\) 7.15620 0.482471
\(221\) 0.307411 0.0206787
\(222\) 4.07674 0.273613
\(223\) 4.00568 0.268240 0.134120 0.990965i \(-0.457179\pi\)
0.134120 + 0.990965i \(0.457179\pi\)
\(224\) 4.90651 0.327830
\(225\) 6.59470 0.439647
\(226\) −6.12533 −0.407451
\(227\) 19.7262 1.30927 0.654636 0.755944i \(-0.272822\pi\)
0.654636 + 0.755944i \(0.272822\pi\)
\(228\) −0.986685 −0.0653448
\(229\) 3.34815 0.221252 0.110626 0.993862i \(-0.464714\pi\)
0.110626 + 0.993862i \(0.464714\pi\)
\(230\) 1.73131 0.114159
\(231\) −24.8465 −1.63478
\(232\) −4.49268 −0.294959
\(233\) −24.7957 −1.62442 −0.812211 0.583364i \(-0.801736\pi\)
−0.812211 + 0.583364i \(0.801736\pi\)
\(234\) 4.73858 0.309770
\(235\) −7.50296 −0.489439
\(236\) −6.28216 −0.408934
\(237\) 3.83464 0.249086
\(238\) 0.504816 0.0327224
\(239\) 14.0875 0.911242 0.455621 0.890174i \(-0.349417\pi\)
0.455621 + 0.890174i \(0.349417\pi\)
\(240\) −6.48067 −0.418325
\(241\) 6.87248 0.442695 0.221348 0.975195i \(-0.428954\pi\)
0.221348 + 0.975195i \(0.428954\pi\)
\(242\) −5.40816 −0.347650
\(243\) 14.9914 0.961697
\(244\) −10.0775 −0.645148
\(245\) 51.6696 3.30105
\(246\) −25.1478 −1.60337
\(247\) 1.37665 0.0875939
\(248\) −1.20673 −0.0766274
\(249\) 17.7691 1.12607
\(250\) −2.54748 −0.161117
\(251\) −21.6696 −1.36777 −0.683887 0.729588i \(-0.739712\pi\)
−0.683887 + 0.729588i \(0.739712\pi\)
\(252\) 7.78147 0.490186
\(253\) 1.35284 0.0850525
\(254\) −3.67448 −0.230558
\(255\) −0.666778 −0.0417552
\(256\) 1.00000 0.0625000
\(257\) 0.662040 0.0412969 0.0206485 0.999787i \(-0.493427\pi\)
0.0206485 + 0.999787i \(0.493427\pi\)
\(258\) −15.3232 −0.953978
\(259\) −9.34050 −0.580391
\(260\) 9.04198 0.560760
\(261\) −7.12517 −0.441037
\(262\) 11.9430 0.737841
\(263\) 10.0437 0.619318 0.309659 0.950848i \(-0.399785\pi\)
0.309659 + 0.950848i \(0.399785\pi\)
\(264\) −5.06398 −0.311666
\(265\) 19.3130 1.18639
\(266\) 2.26066 0.138610
\(267\) −13.7658 −0.842454
\(268\) −0.866688 −0.0529414
\(269\) −24.7612 −1.50972 −0.754858 0.655888i \(-0.772294\pi\)
−0.754858 + 0.655888i \(0.772294\pi\)
\(270\) 9.16399 0.557703
\(271\) −2.37168 −0.144069 −0.0720345 0.997402i \(-0.522949\pi\)
−0.0720345 + 0.997402i \(0.522949\pi\)
\(272\) 0.102887 0.00623845
\(273\) −31.3939 −1.90005
\(274\) −15.3823 −0.929280
\(275\) 9.83294 0.592949
\(276\) −1.22514 −0.0737445
\(277\) −6.96992 −0.418782 −0.209391 0.977832i \(-0.567148\pi\)
−0.209391 + 0.977832i \(0.567148\pi\)
\(278\) −17.3330 −1.03956
\(279\) −1.91381 −0.114577
\(280\) 14.8483 0.887357
\(281\) 18.9864 1.13263 0.566317 0.824187i \(-0.308368\pi\)
0.566317 + 0.824187i \(0.308368\pi\)
\(282\) 5.30936 0.316168
\(283\) 26.5487 1.57816 0.789080 0.614291i \(-0.210558\pi\)
0.789080 + 0.614291i \(0.210558\pi\)
\(284\) −10.2469 −0.608045
\(285\) −2.98596 −0.176873
\(286\) 7.06539 0.417785
\(287\) 57.6179 3.40108
\(288\) 1.58595 0.0934530
\(289\) −16.9894 −0.999377
\(290\) −13.5960 −0.798384
\(291\) 6.86153 0.402230
\(292\) 0.484504 0.0283535
\(293\) 23.5503 1.37582 0.687912 0.725795i \(-0.258528\pi\)
0.687912 + 0.725795i \(0.258528\pi\)
\(294\) −36.5632 −2.13241
\(295\) −19.0114 −1.10689
\(296\) −1.90370 −0.110650
\(297\) 7.16073 0.415507
\(298\) 2.21301 0.128196
\(299\) 1.70934 0.0988536
\(300\) −8.90473 −0.514115
\(301\) 35.1080 2.02359
\(302\) 14.7036 0.846098
\(303\) 34.3215 1.97172
\(304\) 0.460748 0.0264257
\(305\) −30.4971 −1.74626
\(306\) 0.163174 0.00932803
\(307\) −2.33030 −0.132997 −0.0664987 0.997787i \(-0.521183\pi\)
−0.0664987 + 0.997787i \(0.521183\pi\)
\(308\) 11.6025 0.661111
\(309\) 26.0993 1.48474
\(310\) −3.65187 −0.207412
\(311\) 17.5037 0.992541 0.496271 0.868168i \(-0.334702\pi\)
0.496271 + 0.868168i \(0.334702\pi\)
\(312\) −6.39843 −0.362240
\(313\) −25.6728 −1.45111 −0.725557 0.688162i \(-0.758418\pi\)
−0.725557 + 0.688162i \(0.758418\pi\)
\(314\) 11.5889 0.654001
\(315\) 23.5487 1.32682
\(316\) −1.79064 −0.100732
\(317\) −19.5291 −1.09686 −0.548432 0.836195i \(-0.684775\pi\)
−0.548432 + 0.836195i \(0.684775\pi\)
\(318\) −13.6666 −0.766384
\(319\) −10.6239 −0.594823
\(320\) 3.02625 0.169173
\(321\) −34.2705 −1.91279
\(322\) 2.80700 0.156428
\(323\) 0.0474051 0.00263769
\(324\) −11.2426 −0.624590
\(325\) 12.4241 0.689164
\(326\) 4.95134 0.274229
\(327\) 26.7909 1.48154
\(328\) 11.7432 0.648408
\(329\) −12.1647 −0.670659
\(330\) −15.3249 −0.843607
\(331\) 24.5242 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(332\) −8.29759 −0.455389
\(333\) −3.01917 −0.165449
\(334\) 7.93708 0.434298
\(335\) −2.62282 −0.143300
\(336\) −10.5072 −0.573215
\(337\) 11.1072 0.605049 0.302525 0.953142i \(-0.402171\pi\)
0.302525 + 0.953142i \(0.402171\pi\)
\(338\) −4.07277 −0.221529
\(339\) 13.1173 0.712433
\(340\) 0.311363 0.0168860
\(341\) −2.85356 −0.154529
\(342\) 0.730723 0.0395130
\(343\) 49.4271 2.66881
\(344\) 7.15540 0.385793
\(345\) −3.70757 −0.199609
\(346\) 10.4968 0.564310
\(347\) 24.3878 1.30920 0.654602 0.755974i \(-0.272836\pi\)
0.654602 + 0.755974i \(0.272836\pi\)
\(348\) 9.62100 0.515740
\(349\) −5.06912 −0.271344 −0.135672 0.990754i \(-0.543319\pi\)
−0.135672 + 0.990754i \(0.543319\pi\)
\(350\) 20.4023 1.09055
\(351\) 9.04770 0.482930
\(352\) 2.36471 0.126039
\(353\) −20.0993 −1.06978 −0.534888 0.844923i \(-0.679646\pi\)
−0.534888 + 0.844923i \(0.679646\pi\)
\(354\) 13.4531 0.715026
\(355\) −31.0099 −1.64583
\(356\) 6.42816 0.340692
\(357\) −1.08106 −0.0572155
\(358\) −3.74264 −0.197805
\(359\) 24.8402 1.31102 0.655508 0.755188i \(-0.272454\pi\)
0.655508 + 0.755188i \(0.272454\pi\)
\(360\) 4.79948 0.252955
\(361\) −18.7877 −0.988827
\(362\) 10.2544 0.538957
\(363\) 11.5815 0.607870
\(364\) 14.6599 0.768387
\(365\) 1.46623 0.0767461
\(366\) 21.5809 1.12805
\(367\) 27.8619 1.45438 0.727189 0.686437i \(-0.240826\pi\)
0.727189 + 0.686437i \(0.240826\pi\)
\(368\) 0.572097 0.0298226
\(369\) 18.6241 0.969531
\(370\) −5.76107 −0.299504
\(371\) 31.3125 1.62566
\(372\) 2.58419 0.133984
\(373\) −15.6163 −0.808580 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(374\) 0.243298 0.0125806
\(375\) 5.45539 0.281715
\(376\) −2.47929 −0.127860
\(377\) −13.4235 −0.691343
\(378\) 14.8577 0.764198
\(379\) −0.436254 −0.0224089 −0.0112044 0.999937i \(-0.503567\pi\)
−0.0112044 + 0.999937i \(0.503567\pi\)
\(380\) 1.39434 0.0715282
\(381\) 7.86885 0.403133
\(382\) −13.3846 −0.684815
\(383\) 2.41816 0.123562 0.0617811 0.998090i \(-0.480322\pi\)
0.0617811 + 0.998090i \(0.480322\pi\)
\(384\) −2.14148 −0.109282
\(385\) 35.1119 1.78947
\(386\) −13.0787 −0.665689
\(387\) 11.3481 0.576856
\(388\) −3.20410 −0.162664
\(389\) −12.4124 −0.629336 −0.314668 0.949202i \(-0.601893\pi\)
−0.314668 + 0.949202i \(0.601893\pi\)
\(390\) −19.3633 −0.980496
\(391\) 0.0588614 0.00297675
\(392\) 17.0738 0.862357
\(393\) −25.5757 −1.29013
\(394\) 0.464823 0.0234174
\(395\) −5.41894 −0.272657
\(396\) 3.75031 0.188460
\(397\) −18.4432 −0.925639 −0.462819 0.886453i \(-0.653162\pi\)
−0.462819 + 0.886453i \(0.653162\pi\)
\(398\) −15.2055 −0.762181
\(399\) −4.84117 −0.242362
\(400\) 4.15821 0.207910
\(401\) 27.7066 1.38360 0.691801 0.722088i \(-0.256817\pi\)
0.691801 + 0.722088i \(0.256817\pi\)
\(402\) 1.85600 0.0925687
\(403\) −3.60552 −0.179604
\(404\) −16.0270 −0.797373
\(405\) −34.0230 −1.69062
\(406\) −22.0434 −1.09399
\(407\) −4.50169 −0.223140
\(408\) −0.220331 −0.0109080
\(409\) 7.93041 0.392133 0.196067 0.980591i \(-0.437183\pi\)
0.196067 + 0.980591i \(0.437183\pi\)
\(410\) 35.5378 1.75509
\(411\) 32.9410 1.62486
\(412\) −12.1875 −0.600434
\(413\) −30.8234 −1.51672
\(414\) 0.907317 0.0445922
\(415\) −25.1106 −1.23263
\(416\) 2.98785 0.146491
\(417\) 37.1183 1.81769
\(418\) 1.08953 0.0532909
\(419\) 8.49355 0.414937 0.207468 0.978242i \(-0.433478\pi\)
0.207468 + 0.978242i \(0.433478\pi\)
\(420\) −31.7974 −1.55156
\(421\) −29.9497 −1.45966 −0.729829 0.683630i \(-0.760400\pi\)
−0.729829 + 0.683630i \(0.760400\pi\)
\(422\) −7.19882 −0.350433
\(423\) −3.93203 −0.191182
\(424\) 6.38183 0.309929
\(425\) 0.427826 0.0207526
\(426\) 21.9437 1.06317
\(427\) −49.4455 −2.39283
\(428\) 16.0032 0.773542
\(429\) −15.1304 −0.730503
\(430\) 21.6540 1.04425
\(431\) −15.3279 −0.738319 −0.369160 0.929366i \(-0.620354\pi\)
−0.369160 + 0.929366i \(0.620354\pi\)
\(432\) 3.02817 0.145693
\(433\) −1.71995 −0.0826553 −0.0413277 0.999146i \(-0.513159\pi\)
−0.0413277 + 0.999146i \(0.513159\pi\)
\(434\) −5.92082 −0.284209
\(435\) 29.1156 1.39599
\(436\) −12.5104 −0.599142
\(437\) 0.263593 0.0126094
\(438\) −1.03756 −0.0495764
\(439\) −19.7870 −0.944380 −0.472190 0.881497i \(-0.656536\pi\)
−0.472190 + 0.881497i \(0.656536\pi\)
\(440\) 7.15620 0.341159
\(441\) 27.0782 1.28944
\(442\) 0.307411 0.0146221
\(443\) −17.0676 −0.810907 −0.405453 0.914116i \(-0.632886\pi\)
−0.405453 + 0.914116i \(0.632886\pi\)
\(444\) 4.07674 0.193473
\(445\) 19.4533 0.922173
\(446\) 4.00568 0.189675
\(447\) −4.73912 −0.224152
\(448\) 4.90651 0.231811
\(449\) 33.8704 1.59844 0.799221 0.601038i \(-0.205246\pi\)
0.799221 + 0.601038i \(0.205246\pi\)
\(450\) 6.59470 0.310877
\(451\) 27.7692 1.30760
\(452\) −6.12533 −0.288111
\(453\) −31.4876 −1.47941
\(454\) 19.7262 0.925795
\(455\) 44.3645 2.07984
\(456\) −0.986685 −0.0462057
\(457\) 12.5249 0.585890 0.292945 0.956129i \(-0.405365\pi\)
0.292945 + 0.956129i \(0.405365\pi\)
\(458\) 3.34815 0.156449
\(459\) 0.311559 0.0145423
\(460\) 1.73131 0.0807228
\(461\) 15.6533 0.729048 0.364524 0.931194i \(-0.381232\pi\)
0.364524 + 0.931194i \(0.381232\pi\)
\(462\) −24.8465 −1.15596
\(463\) −2.03844 −0.0947344 −0.0473672 0.998878i \(-0.515083\pi\)
−0.0473672 + 0.998878i \(0.515083\pi\)
\(464\) −4.49268 −0.208568
\(465\) 7.82041 0.362663
\(466\) −24.7957 −1.14864
\(467\) 18.8781 0.873574 0.436787 0.899565i \(-0.356116\pi\)
0.436787 + 0.899565i \(0.356116\pi\)
\(468\) 4.73858 0.219041
\(469\) −4.25241 −0.196358
\(470\) −7.50296 −0.346086
\(471\) −24.8175 −1.14353
\(472\) −6.28216 −0.289160
\(473\) 16.9204 0.778002
\(474\) 3.83464 0.176131
\(475\) 1.91589 0.0879069
\(476\) 0.504816 0.0231382
\(477\) 10.1213 0.463421
\(478\) 14.0875 0.644345
\(479\) −17.7215 −0.809714 −0.404857 0.914380i \(-0.632679\pi\)
−0.404857 + 0.914380i \(0.632679\pi\)
\(480\) −6.48067 −0.295801
\(481\) −5.68796 −0.259349
\(482\) 6.87248 0.313033
\(483\) −6.01114 −0.273516
\(484\) −5.40816 −0.245825
\(485\) −9.69642 −0.440292
\(486\) 14.9914 0.680022
\(487\) 13.0609 0.591847 0.295924 0.955212i \(-0.404373\pi\)
0.295924 + 0.955212i \(0.404373\pi\)
\(488\) −10.0775 −0.456188
\(489\) −10.6032 −0.479493
\(490\) 51.6696 2.33420
\(491\) 34.4920 1.55660 0.778301 0.627892i \(-0.216082\pi\)
0.778301 + 0.627892i \(0.216082\pi\)
\(492\) −25.1478 −1.13375
\(493\) −0.462239 −0.0208182
\(494\) 1.37665 0.0619382
\(495\) 11.3494 0.510116
\(496\) −1.20673 −0.0541837
\(497\) −50.2767 −2.25522
\(498\) 17.7691 0.796254
\(499\) 13.7083 0.613667 0.306834 0.951763i \(-0.400730\pi\)
0.306834 + 0.951763i \(0.400730\pi\)
\(500\) −2.54748 −0.113927
\(501\) −16.9971 −0.759375
\(502\) −21.6696 −0.967163
\(503\) 19.3078 0.860893 0.430446 0.902616i \(-0.358356\pi\)
0.430446 + 0.902616i \(0.358356\pi\)
\(504\) 7.78147 0.346614
\(505\) −48.5017 −2.15830
\(506\) 1.35284 0.0601412
\(507\) 8.72176 0.387347
\(508\) −3.67448 −0.163029
\(509\) 31.5399 1.39798 0.698991 0.715130i \(-0.253633\pi\)
0.698991 + 0.715130i \(0.253633\pi\)
\(510\) −0.666778 −0.0295254
\(511\) 2.37722 0.105162
\(512\) 1.00000 0.0441942
\(513\) 1.39522 0.0616006
\(514\) 0.662040 0.0292013
\(515\) −36.8824 −1.62523
\(516\) −15.3232 −0.674564
\(517\) −5.86280 −0.257846
\(518\) −9.34050 −0.410398
\(519\) −22.4787 −0.986704
\(520\) 9.04198 0.396517
\(521\) 5.03364 0.220528 0.110264 0.993902i \(-0.464830\pi\)
0.110264 + 0.993902i \(0.464830\pi\)
\(522\) −7.12517 −0.311860
\(523\) −0.0738998 −0.00323141 −0.00161571 0.999999i \(-0.500514\pi\)
−0.00161571 + 0.999999i \(0.500514\pi\)
\(524\) 11.9430 0.521733
\(525\) −43.6911 −1.90684
\(526\) 10.0437 0.437924
\(527\) −0.124157 −0.00540836
\(528\) −5.06398 −0.220381
\(529\) −22.6727 −0.985770
\(530\) 19.3130 0.838904
\(531\) −9.96318 −0.432365
\(532\) 2.26066 0.0980122
\(533\) 35.0868 1.51978
\(534\) −13.7658 −0.595705
\(535\) 48.4296 2.09379
\(536\) −0.866688 −0.0374352
\(537\) 8.01481 0.345864
\(538\) −24.7612 −1.06753
\(539\) 40.3745 1.73905
\(540\) 9.16399 0.394355
\(541\) 1.21097 0.0520636 0.0260318 0.999661i \(-0.491713\pi\)
0.0260318 + 0.999661i \(0.491713\pi\)
\(542\) −2.37168 −0.101872
\(543\) −21.9595 −0.942374
\(544\) 0.102887 0.00441125
\(545\) −37.8598 −1.62173
\(546\) −31.3939 −1.34354
\(547\) 4.78244 0.204482 0.102241 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(548\) −15.3823 −0.657101
\(549\) −15.9825 −0.682114
\(550\) 9.83294 0.419278
\(551\) −2.07000 −0.0881848
\(552\) −1.22514 −0.0521453
\(553\) −8.78581 −0.373611
\(554\) −6.96992 −0.296124
\(555\) 12.3372 0.523686
\(556\) −17.3330 −0.735083
\(557\) 9.18550 0.389202 0.194601 0.980882i \(-0.437659\pi\)
0.194601 + 0.980882i \(0.437659\pi\)
\(558\) −1.91381 −0.0810181
\(559\) 21.3792 0.904245
\(560\) 14.8483 0.627456
\(561\) −0.521019 −0.0219974
\(562\) 18.9864 0.800893
\(563\) 14.2454 0.600373 0.300186 0.953881i \(-0.402951\pi\)
0.300186 + 0.953881i \(0.402951\pi\)
\(564\) 5.30936 0.223564
\(565\) −18.5368 −0.779849
\(566\) 26.5487 1.11593
\(567\) −55.1619 −2.31658
\(568\) −10.2469 −0.429952
\(569\) −33.7678 −1.41562 −0.707811 0.706402i \(-0.750317\pi\)
−0.707811 + 0.706402i \(0.750317\pi\)
\(570\) −2.98596 −0.125068
\(571\) −2.55408 −0.106885 −0.0534424 0.998571i \(-0.517019\pi\)
−0.0534424 + 0.998571i \(0.517019\pi\)
\(572\) 7.06539 0.295419
\(573\) 28.6628 1.19741
\(574\) 57.6179 2.40493
\(575\) 2.37890 0.0992069
\(576\) 1.58595 0.0660812
\(577\) 2.53739 0.105633 0.0528164 0.998604i \(-0.483180\pi\)
0.0528164 + 0.998604i \(0.483180\pi\)
\(578\) −16.9894 −0.706666
\(579\) 28.0079 1.16397
\(580\) −13.5960 −0.564543
\(581\) −40.7122 −1.68903
\(582\) 6.86153 0.284420
\(583\) 15.0912 0.625012
\(584\) 0.484504 0.0200489
\(585\) 14.3401 0.592891
\(586\) 23.5503 0.972854
\(587\) −45.3043 −1.86991 −0.934955 0.354766i \(-0.884561\pi\)
−0.934955 + 0.354766i \(0.884561\pi\)
\(588\) −36.5632 −1.50784
\(589\) −0.555998 −0.0229095
\(590\) −19.0114 −0.782687
\(591\) −0.995411 −0.0409457
\(592\) −1.90370 −0.0782415
\(593\) −30.1819 −1.23942 −0.619710 0.784831i \(-0.712750\pi\)
−0.619710 + 0.784831i \(0.712750\pi\)
\(594\) 7.16073 0.293808
\(595\) 1.52770 0.0626297
\(596\) 2.21301 0.0906483
\(597\) 32.5622 1.33268
\(598\) 1.70934 0.0699001
\(599\) 3.49153 0.142660 0.0713300 0.997453i \(-0.477276\pi\)
0.0713300 + 0.997453i \(0.477276\pi\)
\(600\) −8.90473 −0.363534
\(601\) −2.92013 −0.119115 −0.0595573 0.998225i \(-0.518969\pi\)
−0.0595573 + 0.998225i \(0.518969\pi\)
\(602\) 35.1080 1.43089
\(603\) −1.37452 −0.0559749
\(604\) 14.7036 0.598282
\(605\) −16.3665 −0.665391
\(606\) 34.3215 1.39422
\(607\) 21.9260 0.889950 0.444975 0.895543i \(-0.353213\pi\)
0.444975 + 0.895543i \(0.353213\pi\)
\(608\) 0.460748 0.0186858
\(609\) 47.2055 1.91286
\(610\) −30.4971 −1.23479
\(611\) −7.40775 −0.299685
\(612\) 0.163174 0.00659591
\(613\) 19.2838 0.778864 0.389432 0.921055i \(-0.372671\pi\)
0.389432 + 0.921055i \(0.372671\pi\)
\(614\) −2.33030 −0.0940434
\(615\) −76.1036 −3.06879
\(616\) 11.6025 0.467476
\(617\) 38.9538 1.56822 0.784110 0.620622i \(-0.213120\pi\)
0.784110 + 0.620622i \(0.213120\pi\)
\(618\) 26.0993 1.04987
\(619\) −23.1883 −0.932018 −0.466009 0.884780i \(-0.654309\pi\)
−0.466009 + 0.884780i \(0.654309\pi\)
\(620\) −3.65187 −0.146663
\(621\) 1.73240 0.0695190
\(622\) 17.5037 0.701833
\(623\) 31.5398 1.26362
\(624\) −6.39843 −0.256142
\(625\) −28.5004 −1.14001
\(626\) −25.6728 −1.02609
\(627\) −2.33322 −0.0931798
\(628\) 11.5889 0.462449
\(629\) −0.195866 −0.00780969
\(630\) 23.5487 0.938202
\(631\) 21.4699 0.854704 0.427352 0.904085i \(-0.359447\pi\)
0.427352 + 0.904085i \(0.359447\pi\)
\(632\) −1.79064 −0.0712280
\(633\) 15.4161 0.612737
\(634\) −19.5291 −0.775599
\(635\) −11.1199 −0.441280
\(636\) −13.6666 −0.541915
\(637\) 51.0139 2.02124
\(638\) −10.6239 −0.420603
\(639\) −16.2511 −0.642885
\(640\) 3.02625 0.119623
\(641\) 10.4306 0.411985 0.205993 0.978554i \(-0.433958\pi\)
0.205993 + 0.978554i \(0.433958\pi\)
\(642\) −34.2705 −1.35255
\(643\) −20.5756 −0.811422 −0.405711 0.914001i \(-0.632976\pi\)
−0.405711 + 0.914001i \(0.632976\pi\)
\(644\) 2.80700 0.110611
\(645\) −46.3717 −1.82589
\(646\) 0.0474051 0.00186513
\(647\) 34.3624 1.35092 0.675462 0.737395i \(-0.263944\pi\)
0.675462 + 0.737395i \(0.263944\pi\)
\(648\) −11.2426 −0.441652
\(649\) −14.8555 −0.583128
\(650\) 12.4241 0.487313
\(651\) 12.6793 0.496943
\(652\) 4.95134 0.193909
\(653\) −5.38313 −0.210658 −0.105329 0.994437i \(-0.533590\pi\)
−0.105329 + 0.994437i \(0.533590\pi\)
\(654\) 26.7909 1.04761
\(655\) 36.1426 1.41221
\(656\) 11.7432 0.458494
\(657\) 0.768400 0.0299781
\(658\) −12.1647 −0.474228
\(659\) −20.5577 −0.800815 −0.400407 0.916337i \(-0.631131\pi\)
−0.400407 + 0.916337i \(0.631131\pi\)
\(660\) −15.3249 −0.596520
\(661\) 8.53375 0.331924 0.165962 0.986132i \(-0.446927\pi\)
0.165962 + 0.986132i \(0.446927\pi\)
\(662\) 24.5242 0.953162
\(663\) −0.658316 −0.0255669
\(664\) −8.29759 −0.322009
\(665\) 6.84134 0.265296
\(666\) −3.01917 −0.116990
\(667\) −2.57025 −0.0995205
\(668\) 7.93708 0.307095
\(669\) −8.57810 −0.331648
\(670\) −2.62282 −0.101328
\(671\) −23.8304 −0.919963
\(672\) −10.5072 −0.405324
\(673\) 14.7422 0.568269 0.284135 0.958784i \(-0.408294\pi\)
0.284135 + 0.958784i \(0.408294\pi\)
\(674\) 11.1072 0.427835
\(675\) 12.5917 0.484656
\(676\) −4.07277 −0.156645
\(677\) −24.5160 −0.942228 −0.471114 0.882072i \(-0.656148\pi\)
−0.471114 + 0.882072i \(0.656148\pi\)
\(678\) 13.1173 0.503766
\(679\) −15.7209 −0.603314
\(680\) 0.311363 0.0119402
\(681\) −42.2432 −1.61876
\(682\) −2.85356 −0.109269
\(683\) −17.0194 −0.651229 −0.325614 0.945503i \(-0.605571\pi\)
−0.325614 + 0.945503i \(0.605571\pi\)
\(684\) 0.730723 0.0279399
\(685\) −46.5508 −1.77862
\(686\) 49.4271 1.88714
\(687\) −7.17000 −0.273553
\(688\) 7.15540 0.272797
\(689\) 19.0679 0.726430
\(690\) −3.70757 −0.141145
\(691\) 27.9825 1.06450 0.532251 0.846586i \(-0.321346\pi\)
0.532251 + 0.846586i \(0.321346\pi\)
\(692\) 10.4968 0.399028
\(693\) 18.4009 0.698992
\(694\) 24.3878 0.925747
\(695\) −52.4540 −1.98970
\(696\) 9.62100 0.364683
\(697\) 1.20822 0.0457647
\(698\) −5.06912 −0.191869
\(699\) 53.0996 2.00841
\(700\) 20.4023 0.771133
\(701\) −23.4286 −0.884886 −0.442443 0.896797i \(-0.645888\pi\)
−0.442443 + 0.896797i \(0.645888\pi\)
\(702\) 9.04770 0.341483
\(703\) −0.877125 −0.0330814
\(704\) 2.36471 0.0891233
\(705\) 16.0675 0.605136
\(706\) −20.0993 −0.756446
\(707\) −78.6365 −2.95743
\(708\) 13.4531 0.505600
\(709\) 11.2857 0.423842 0.211921 0.977287i \(-0.432028\pi\)
0.211921 + 0.977287i \(0.432028\pi\)
\(710\) −31.0099 −1.16378
\(711\) −2.83987 −0.106504
\(712\) 6.42816 0.240906
\(713\) −0.690366 −0.0258544
\(714\) −1.08106 −0.0404575
\(715\) 21.3816 0.799628
\(716\) −3.74264 −0.139869
\(717\) −30.1680 −1.12665
\(718\) 24.8402 0.927029
\(719\) −36.1228 −1.34715 −0.673577 0.739117i \(-0.735243\pi\)
−0.673577 + 0.739117i \(0.735243\pi\)
\(720\) 4.79948 0.178866
\(721\) −59.7979 −2.22699
\(722\) −18.7877 −0.699206
\(723\) −14.7173 −0.547342
\(724\) 10.2544 0.381100
\(725\) −18.6815 −0.693813
\(726\) 11.5815 0.429829
\(727\) −30.6948 −1.13841 −0.569204 0.822196i \(-0.692749\pi\)
−0.569204 + 0.822196i \(0.692749\pi\)
\(728\) 14.6599 0.543332
\(729\) 1.62408 0.0601511
\(730\) 1.46623 0.0542677
\(731\) 0.736198 0.0272293
\(732\) 21.5809 0.797651
\(733\) −11.5334 −0.425995 −0.212997 0.977053i \(-0.568323\pi\)
−0.212997 + 0.977053i \(0.568323\pi\)
\(734\) 27.8619 1.02840
\(735\) −110.650 −4.08137
\(736\) 0.572097 0.0210878
\(737\) −2.04946 −0.0754930
\(738\) 18.6241 0.685562
\(739\) 11.9344 0.439015 0.219507 0.975611i \(-0.429555\pi\)
0.219507 + 0.975611i \(0.429555\pi\)
\(740\) −5.76107 −0.211781
\(741\) −2.94806 −0.108300
\(742\) 31.3125 1.14952
\(743\) 32.9627 1.20928 0.604642 0.796497i \(-0.293316\pi\)
0.604642 + 0.796497i \(0.293316\pi\)
\(744\) 2.58419 0.0947410
\(745\) 6.69712 0.245363
\(746\) −15.6163 −0.571752
\(747\) −13.1596 −0.481483
\(748\) 0.243298 0.00889586
\(749\) 78.5196 2.86904
\(750\) 5.45539 0.199203
\(751\) −33.4445 −1.22041 −0.610204 0.792244i \(-0.708913\pi\)
−0.610204 + 0.792244i \(0.708913\pi\)
\(752\) −2.47929 −0.0904105
\(753\) 46.4051 1.69110
\(754\) −13.4235 −0.488853
\(755\) 44.4969 1.61941
\(756\) 14.8577 0.540370
\(757\) 34.4527 1.25220 0.626102 0.779741i \(-0.284649\pi\)
0.626102 + 0.779741i \(0.284649\pi\)
\(758\) −0.436254 −0.0158455
\(759\) −2.89709 −0.105158
\(760\) 1.39434 0.0505780
\(761\) −30.8397 −1.11794 −0.558970 0.829188i \(-0.688803\pi\)
−0.558970 + 0.829188i \(0.688803\pi\)
\(762\) 7.86885 0.285058
\(763\) −61.3826 −2.22220
\(764\) −13.3846 −0.484237
\(765\) 0.493805 0.0178536
\(766\) 2.41816 0.0873716
\(767\) −18.7701 −0.677750
\(768\) −2.14148 −0.0772741
\(769\) 11.3229 0.408315 0.204157 0.978938i \(-0.434555\pi\)
0.204157 + 0.978938i \(0.434555\pi\)
\(770\) 35.1119 1.26535
\(771\) −1.41775 −0.0510589
\(772\) −13.0787 −0.470713
\(773\) 14.4054 0.518125 0.259063 0.965861i \(-0.416586\pi\)
0.259063 + 0.965861i \(0.416586\pi\)
\(774\) 11.3481 0.407899
\(775\) −5.01783 −0.180246
\(776\) −3.20410 −0.115021
\(777\) 20.0025 0.717587
\(778\) −12.4124 −0.445008
\(779\) 5.41064 0.193856
\(780\) −19.3633 −0.693316
\(781\) −24.2310 −0.867055
\(782\) 0.0588614 0.00210488
\(783\) −13.6046 −0.486188
\(784\) 17.0738 0.609778
\(785\) 35.0710 1.25174
\(786\) −25.5757 −0.912257
\(787\) 12.1680 0.433742 0.216871 0.976200i \(-0.430415\pi\)
0.216871 + 0.976200i \(0.430415\pi\)
\(788\) 0.464823 0.0165586
\(789\) −21.5083 −0.765716
\(790\) −5.41894 −0.192797
\(791\) −30.0540 −1.06860
\(792\) 3.75031 0.133261
\(793\) −30.1101 −1.06924
\(794\) −18.4432 −0.654525
\(795\) −41.3585 −1.46684
\(796\) −15.2055 −0.538943
\(797\) 26.1475 0.926193 0.463097 0.886308i \(-0.346738\pi\)
0.463097 + 0.886308i \(0.346738\pi\)
\(798\) −4.84117 −0.171376
\(799\) −0.255087 −0.00902434
\(800\) 4.15821 0.147015
\(801\) 10.1947 0.360214
\(802\) 27.7066 0.978355
\(803\) 1.14571 0.0404313
\(804\) 1.85600 0.0654560
\(805\) 8.49468 0.299398
\(806\) −3.60552 −0.126999
\(807\) 53.0257 1.86659
\(808\) −16.0270 −0.563828
\(809\) −52.7718 −1.85536 −0.927678 0.373381i \(-0.878199\pi\)
−0.927678 + 0.373381i \(0.878199\pi\)
\(810\) −34.0230 −1.19545
\(811\) −33.7966 −1.18676 −0.593380 0.804922i \(-0.702207\pi\)
−0.593380 + 0.804922i \(0.702207\pi\)
\(812\) −22.0434 −0.773571
\(813\) 5.07890 0.178125
\(814\) −4.50169 −0.157784
\(815\) 14.9840 0.524866
\(816\) −0.220331 −0.00771313
\(817\) 3.29684 0.115342
\(818\) 7.93041 0.277280
\(819\) 23.2498 0.812415
\(820\) 35.5378 1.24103
\(821\) −38.9190 −1.35828 −0.679142 0.734007i \(-0.737648\pi\)
−0.679142 + 0.734007i \(0.737648\pi\)
\(822\) 32.9410 1.14895
\(823\) 21.3175 0.743083 0.371541 0.928416i \(-0.378829\pi\)
0.371541 + 0.928416i \(0.378829\pi\)
\(824\) −12.1875 −0.424571
\(825\) −21.0571 −0.733113
\(826\) −30.8234 −1.07248
\(827\) 40.3624 1.40354 0.701768 0.712405i \(-0.252394\pi\)
0.701768 + 0.712405i \(0.252394\pi\)
\(828\) 0.907317 0.0315314
\(829\) −20.5576 −0.713994 −0.356997 0.934105i \(-0.616199\pi\)
−0.356997 + 0.934105i \(0.616199\pi\)
\(830\) −25.1106 −0.871602
\(831\) 14.9260 0.517776
\(832\) 2.98785 0.103585
\(833\) 1.75667 0.0608652
\(834\) 37.1183 1.28530
\(835\) 24.0196 0.831233
\(836\) 1.08953 0.0376824
\(837\) −3.65418 −0.126307
\(838\) 8.49355 0.293405
\(839\) 6.90658 0.238441 0.119221 0.992868i \(-0.461960\pi\)
0.119221 + 0.992868i \(0.461960\pi\)
\(840\) −31.7974 −1.09712
\(841\) −8.81580 −0.303993
\(842\) −29.9497 −1.03213
\(843\) −40.6591 −1.40037
\(844\) −7.19882 −0.247793
\(845\) −12.3252 −0.424000
\(846\) −3.93203 −0.135186
\(847\) −26.5352 −0.911759
\(848\) 6.38183 0.219153
\(849\) −56.8537 −1.95121
\(850\) 0.427826 0.0146743
\(851\) −1.08910 −0.0373339
\(852\) 21.9437 0.751778
\(853\) −38.0363 −1.30234 −0.651169 0.758933i \(-0.725721\pi\)
−0.651169 + 0.758933i \(0.725721\pi\)
\(854\) −49.4455 −1.69199
\(855\) 2.21135 0.0756267
\(856\) 16.0032 0.546977
\(857\) 29.9318 1.02245 0.511226 0.859446i \(-0.329191\pi\)
0.511226 + 0.859446i \(0.329191\pi\)
\(858\) −15.1304 −0.516544
\(859\) 48.1047 1.64131 0.820655 0.571424i \(-0.193609\pi\)
0.820655 + 0.571424i \(0.193609\pi\)
\(860\) 21.6540 0.738396
\(861\) −123.388 −4.20505
\(862\) −15.3279 −0.522071
\(863\) 42.5551 1.44859 0.724296 0.689489i \(-0.242165\pi\)
0.724296 + 0.689489i \(0.242165\pi\)
\(864\) 3.02817 0.103020
\(865\) 31.7659 1.08007
\(866\) −1.71995 −0.0584461
\(867\) 36.3825 1.23562
\(868\) −5.92082 −0.200966
\(869\) −4.23435 −0.143641
\(870\) 29.1156 0.987111
\(871\) −2.58953 −0.0877429
\(872\) −12.5104 −0.423657
\(873\) −5.08154 −0.171984
\(874\) 0.263593 0.00891616
\(875\) −12.4992 −0.422552
\(876\) −1.03756 −0.0350558
\(877\) 19.5922 0.661580 0.330790 0.943704i \(-0.392685\pi\)
0.330790 + 0.943704i \(0.392685\pi\)
\(878\) −19.7870 −0.667778
\(879\) −50.4326 −1.70105
\(880\) 7.15620 0.241236
\(881\) 8.11407 0.273370 0.136685 0.990615i \(-0.456355\pi\)
0.136685 + 0.990615i \(0.456355\pi\)
\(882\) 27.0782 0.911770
\(883\) −43.6962 −1.47049 −0.735247 0.677800i \(-0.762934\pi\)
−0.735247 + 0.677800i \(0.762934\pi\)
\(884\) 0.307411 0.0103394
\(885\) 40.7126 1.36854
\(886\) −17.0676 −0.573398
\(887\) −9.72075 −0.326391 −0.163195 0.986594i \(-0.552180\pi\)
−0.163195 + 0.986594i \(0.552180\pi\)
\(888\) 4.07674 0.136806
\(889\) −18.0289 −0.604669
\(890\) 19.4533 0.652074
\(891\) −26.5855 −0.890647
\(892\) 4.00568 0.134120
\(893\) −1.14233 −0.0382266
\(894\) −4.73912 −0.158500
\(895\) −11.3262 −0.378593
\(896\) 4.90651 0.163915
\(897\) −3.66052 −0.122221
\(898\) 33.8704 1.13027
\(899\) 5.42145 0.180816
\(900\) 6.59470 0.219823
\(901\) 0.656608 0.0218748
\(902\) 27.7692 0.924612
\(903\) −75.1832 −2.50194
\(904\) −6.12533 −0.203725
\(905\) 31.0323 1.03155
\(906\) −31.4876 −1.04610
\(907\) 8.94554 0.297032 0.148516 0.988910i \(-0.452550\pi\)
0.148516 + 0.988910i \(0.452550\pi\)
\(908\) 19.7262 0.654636
\(909\) −25.4180 −0.843062
\(910\) 44.3645 1.47067
\(911\) 54.8206 1.81629 0.908144 0.418658i \(-0.137499\pi\)
0.908144 + 0.418658i \(0.137499\pi\)
\(912\) −0.986685 −0.0326724
\(913\) −19.6214 −0.649373
\(914\) 12.5249 0.414286
\(915\) 65.3091 2.15905
\(916\) 3.34815 0.110626
\(917\) 58.5984 1.93509
\(918\) 0.311559 0.0102830
\(919\) 54.8896 1.81064 0.905320 0.424729i \(-0.139631\pi\)
0.905320 + 0.424729i \(0.139631\pi\)
\(920\) 1.73131 0.0570796
\(921\) 4.99030 0.164436
\(922\) 15.6533 0.515515
\(923\) −30.6163 −1.00775
\(924\) −24.8465 −0.817388
\(925\) −7.91596 −0.260275
\(926\) −2.03844 −0.0669873
\(927\) −19.3287 −0.634839
\(928\) −4.49268 −0.147480
\(929\) 51.8191 1.70013 0.850065 0.526677i \(-0.176562\pi\)
0.850065 + 0.526677i \(0.176562\pi\)
\(930\) 7.82041 0.256441
\(931\) 7.86672 0.257821
\(932\) −24.7957 −0.812211
\(933\) −37.4838 −1.22716
\(934\) 18.8781 0.617710
\(935\) 0.736281 0.0240790
\(936\) 4.73858 0.154885
\(937\) 30.6458 1.00115 0.500577 0.865692i \(-0.333121\pi\)
0.500577 + 0.865692i \(0.333121\pi\)
\(938\) −4.25241 −0.138846
\(939\) 54.9779 1.79414
\(940\) −7.50296 −0.244720
\(941\) −39.6461 −1.29243 −0.646214 0.763157i \(-0.723648\pi\)
−0.646214 + 0.763157i \(0.723648\pi\)
\(942\) −24.8175 −0.808598
\(943\) 6.71823 0.218776
\(944\) −6.28216 −0.204467
\(945\) 44.9632 1.46265
\(946\) 16.9204 0.550130
\(947\) −17.3138 −0.562624 −0.281312 0.959616i \(-0.590770\pi\)
−0.281312 + 0.959616i \(0.590770\pi\)
\(948\) 3.83464 0.124543
\(949\) 1.44763 0.0469919
\(950\) 1.91589 0.0621595
\(951\) 41.8212 1.35615
\(952\) 0.504816 0.0163612
\(953\) −5.11114 −0.165566 −0.0827831 0.996568i \(-0.526381\pi\)
−0.0827831 + 0.996568i \(0.526381\pi\)
\(954\) 10.1213 0.327688
\(955\) −40.5051 −1.31071
\(956\) 14.0875 0.455621
\(957\) 22.7509 0.735431
\(958\) −17.7215 −0.572554
\(959\) −75.4735 −2.43717
\(960\) −6.48067 −0.209163
\(961\) −29.5438 −0.953026
\(962\) −5.68796 −0.183387
\(963\) 25.3802 0.817866
\(964\) 6.87248 0.221348
\(965\) −39.5795 −1.27411
\(966\) −6.01114 −0.193405
\(967\) −30.5212 −0.981497 −0.490748 0.871301i \(-0.663277\pi\)
−0.490748 + 0.871301i \(0.663277\pi\)
\(968\) −5.40816 −0.173825
\(969\) −0.101517 −0.00326120
\(970\) −9.69642 −0.311333
\(971\) −18.4468 −0.591985 −0.295993 0.955190i \(-0.595650\pi\)
−0.295993 + 0.955190i \(0.595650\pi\)
\(972\) 14.9914 0.480848
\(973\) −85.0445 −2.72640
\(974\) 13.0609 0.418499
\(975\) −26.6060 −0.852073
\(976\) −10.0775 −0.322574
\(977\) −34.6632 −1.10897 −0.554487 0.832193i \(-0.687085\pi\)
−0.554487 + 0.832193i \(0.687085\pi\)
\(978\) −10.6032 −0.339053
\(979\) 15.2007 0.485817
\(980\) 51.6696 1.65053
\(981\) −19.8409 −0.633472
\(982\) 34.4920 1.10068
\(983\) −9.69657 −0.309272 −0.154636 0.987971i \(-0.549421\pi\)
−0.154636 + 0.987971i \(0.549421\pi\)
\(984\) −25.1478 −0.801683
\(985\) 1.40667 0.0448203
\(986\) −0.462239 −0.0147207
\(987\) 26.0504 0.829194
\(988\) 1.37665 0.0437969
\(989\) 4.09358 0.130168
\(990\) 11.3494 0.360707
\(991\) −16.9811 −0.539423 −0.269711 0.962941i \(-0.586928\pi\)
−0.269711 + 0.962941i \(0.586928\pi\)
\(992\) −1.20673 −0.0383137
\(993\) −52.5183 −1.66662
\(994\) −50.2767 −1.59468
\(995\) −46.0156 −1.45879
\(996\) 17.7691 0.563037
\(997\) 8.77329 0.277853 0.138926 0.990303i \(-0.455635\pi\)
0.138926 + 0.990303i \(0.455635\pi\)
\(998\) 13.7083 0.433928
\(999\) −5.76471 −0.182387
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6046.2.a.f.1.12 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.f.1.12 67 1.1 even 1 trivial